Symmetry Group P2

There are only 17 methods to divide the two dimensional surface into true tessellations. These 17 rules were derived from the observation and study of crystallography, a long time ago.  They are the basis for many other disciplines as well, wallpaper designs, fabric design, wrapping paper, ceramic tiles… But my passion lies with classic tessellations, where the same odd shapes repeat to fill a surface with an absolutely tight perfect fit. Well, I try, anyhow.

The Tofino Hitchhiker was built using this P2 symmetry system.

Symmetry Group P2 – Tofino Hitchhiker – © 2013 Champagne Design

Symmetry Group P2 – Tofino Hitchhiker – © 2013 Champagne Design

Symmetry Group P2

I’ve come to call this symmetry group the “triple whip”. The system is based on a four sided grid, which can be skewed into lozenges, parallelograms. Each of the corners has a two-fold rotation point. One of them, slightly special. If three lines are drawn from that corner, each to one other corner, a perfectly repeating shape is formed. I look at these, I’ve drawn them, but can never quite put it into words as well. The triple whip drawing below, explains it better. Note, rotation point R2, has the 3 whips extending to the other 3 corners. No mirrors or glides here, so your lines can wander over the parallelogram’s delineation.

Symmetry Group P2, explained

Symmetry Group P2, explained

The Hiker image below, when zoomed-out, plainly shows the parallelogram, helped by the horizontal belt and hiking pole of course. Rotation points occur at his right hand, left SO4 watch, top of one boot and bottom of the other. See the 6 red dots. Two of the parallelograms are required to translate the full figure. I’ve read another explanation, where a parallelogram has 4 two-fold rotation points, midway on each side and a fifth two-fold rotation in the middle. But that just confuses me.

Symmetry Group P2 - © 2013 Champagne Design, Vancouver Island Hiker

Symmetry Group P2 – © 2013 Champagne Design, Vancouver Island Hiker

Twelve symmetry groups explained and illustrated to date. I have a bit more work to do, 5 more groups. Till then,

Lots of software out there to help you accomplish this type of design.

  1. An iPad app is available, which is what I have used here to create these images: KaleidoPaint by Jeff Weeks.
  2. There is also a java-based program “Escher Web Sketch”  at the Ecole Polytechnique de Lausanne. Make sure Java is enabled and not blocked by your security software.
  3. Also, another screen-based software by Anselm Levskaya Escher Sketch v2.
  4. Or a pair of scissors and a piece of cardboard works quite well. That’s how I learned.

Comments are always welcome!


If you’re a “Learn by Seeing” “Learn by Doing” kind of person, I’ve started creating videos on “how-to” create tessellations. I’ll be covering each of the 17 symmetry groups, one class at a time. And like all artists, we need to make a living. So. I’ve uploaded these to the Skillshare platform. I’ll get paid by minutes watched.

You can take the classes for free. Skillshare offers anywhere from 2 weeks to a month for free if you sign up, even temporarily.

You can register for just a month and cancel anytime. It’s less than the cost of a Netflix subscription!
There are over 40,000 classes on topics for creative persons just like you. Join my mailing list, either here on my blog (in the sidebar), or a at this link for a specific list I use to announce new classes.

I’d love for you to join me on this wonderful learning adventure.

If you prefer, you can follow my progress on social media, I always announce my new class:
Facebook: Franc Champagne, and Vancouver Island Tessellation Artist
Instagram: champagne.francine
Twitter: FChampagne1
Linkedin: Graphic Design, PowerPoint and tessellations
Youtube: Video animations and class intros


2 thoughts on “Symmetry Group P2

  1. Pingback: Symmetry Group P2 | champagnedesign

  2. Pingback: Four Snails #Tessellation | Vancouver Island Tessellation Artist

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.